Asymptotic Properties of the Wald-Wolfowitz Test of Randomness

Abstract

The paper investigates certain asymptotic properties of the test of randomness based on the statistic $R_h = \sum^n_{i=1} x_ix_{i+h}$ proposed by Wald and Wolfowitz. It is shown that the conditions given in the original paper for asymptotic normality of $R_h$ when the null hypothesis of randomness is true can be weakened considerably. Conditions are given for the consistency of the test when under the alternative hypothesis consecutive observations are drawn independently from changing populations with continuous cumulative distribution functions. In particular a downward (upward) trend and a regular cyclical movement are considered. For the special case of a regular cyclical movement of known length the asymptotic relative efficiency of the test based on ranks with respect to the test based on original observations is found. A simple condition for the asymptotic normality of $R_h$ for ranks under the alternative hypothesis is given. This asymptotic normality is used to compare the asymptotic power of the $R_h$-test with that of the Mann $T$-test in the case of a downward trend.